The survey under review is concentrated around Serre’s Conjecture II stating that every principal homogeneous space of a semisimple simply connected linear algebraic group defined over a perfect field of cohomological dimension two has a rational point. The author, who contributed much to the development of this area of research, gives a comprehensive account of relevant results and methods, not limited by exposition of her own achievements. Putting the historical framework between the Merkurjev-Suslin theorem of the early 1980’s and recent patching techniques of Harbater-Hartmann-Krashen of the late 2000’s, she presents an overview of a variety of particular topics. Here are some of those: works of Bayer-Fluckiger and the author on classical groups; Serre’s treatment of \(G_2\) and \(F_4\); the real analogue of Conjecture II proposed by Colliot-Thélène; use of the Rost invariant for proofs of the original conjecture and its strengthened version; works of Gille, Garibaldi and Chernousov on exceptional groups; arithmetic of linear algebraic groups over some two-dimensional fields.
For the entire collection see [Zbl 1220.00031].